# Logic

Greg Restall
Pages: 240
https://www.jstor.org/stable/j.ctt80f93

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Acknowledgements
(pp. xiii-xiv)
Greg Restall
4. Introduction
(pp. 1-6)
Dorothy Grover and Nuel Belnap

There are many different reasons to study logic. Logic is the theory ofgood reasoning.Studying logic not only helps you to reason well, but it also helps youunderstandhow reasoning works.

Logic can be done in two ways – it can beformaland it can bephilosophical.This book concentrates onbothaspects of logic. So, we’ll be examining the techniques that logicians use inmodellinggood reasoning. This ‘modelling’ is formal and technical, just like the formal modelling you see in other disciplines, such as the physical and social sciences and economics.

The philosophical aspects of logic...

5. PART 1 Prepositional logic
• Chapter 1 Propositions and arguments
(pp. 9-19)

Logic is all aboutreasons.Every day we consider possibilities, we think about what follows from different assumptions, what would be the case in different alternatives, and we weigh up competing positions or options. In all of this, wereason.Logic is the study of good reasoning, and in particular, what makes good reasoninggood.

To understand good reasoning, we must have an idea of the kinds of things we reason about. What are the things we give reasons for? We can give reasons for doing something rather than something else (these are reasons foractions)or for liking some...

• Chapter 2 Connectives and argument forms
(pp. 20-35)

As you saw in the last chapter, arguments have different forms, and we can use forms of arguments in our study of validity and invalidity of arguments. The forms of an argument bring to light a kind ofstructureexhibited by the argument. Different sorts of forms expose different sorts of structure. The first kind of structure we will examine is the structure given byprepositional connectives.These connectives give us ways to construct new propositions out of old propositions. The connectives form the ‘nuts and bolts’ in many different argument forms. The resulting theory is calledpropositional logicand...

• Chapter 3 Truth tables
(pp. 36-54)

In the last chapter, we introduced a formal language, to describe the structure of arguments. We will now use this formal language to analyse argument forms. To do this, we will examine how each connective interacts with truth and falsity, since the truth and falsity of the premises and conclusions of arguments are involved so intimately with the validity of arguments.

We can now go some way in finding how the connectives work in arguments, by noting that propositions are the sort of things that can be true, or false. And the truth value of a complex proposition depends crucially...

• Chapter 4 Trees
(pp. 55-76)

Although the method of assigning values is a great improvement on truth tables, it too can get out of control. It often is difficult to keep track of what is going on, and you don’t get too much of an idea of why the argument is valid or invalid – all you get is one counterexample, if there is one. We will introduce another method for evaluating arguments that is as quick and efficient as the method of assigning values, but is much easier to handle, and that gives you more information about the argument form you are considering. The structures...

• Chapter 5 Vagueness and bivalence
(pp. 77-87)

The theory of logical consequence constructed so far is powerful and useful. It is elegant in its simplicity, but far-reaching in its power and its breadth. There are also some important problems for interpreting and using this account. In this chapter, we will look at one class of problems, which stem from the assumption that each proposition is assigned either the value ‘true’ or the value ‘false’. This is the doctrine of bivalence (‘bivalent’ means ‘has two values’). The first problem stems from the vagueness inherent in our use of language.

Consider a long strip, shading continuously from scarlet at...

• Chapter 6 Conditionality
(pp. 88-101)

Whenever you first see truth tables, one connective is more troublesome than any of the others: implication. The conditional$p \supset q$is false only when p is true and q is false. This does not seem to fit with the way that we use ‘if’. What can be said about this?

The validity of argument forms like these

$p\Pi q \supset p$$p\Pi \sim p \supset q$

follows immediately from the truth table rules for$\supset$. The problem with these argument forms is that they seem to have manyinvalidinstances. Consider the argument from p to$q \supset p$. One instance is the inference fromI am alive. The premise...

• Chapter 7 Natural deduction
(pp. 102-110)

Before turning toPredicate Logicin the second part of the book, we will look at one more way to present the logic of propositions. A system of natural deduction gives you a way to develop proofs of formulas, from basic proofs that are known to be valid.

The rules tell us how to build up complex arguments frombasicarguments. The basic arguments are simple. They are of the form

$X \bot A$

whenever A is a member of the set X. We write sets of formulas by listing their members. So$A \bot A$and A, B,$C \bot B$are two examples.

To...

6. PART 2 Predicate logic
• Chapter 8 Predicates, names and quantifiers
(pp. 113-127)

Prepositional logic, which you know well by now, can establish the validity of many argument forms. As such, it is a very useful tool. However, some arguments are obviously valid, and yet cannot be shown to be so by the methods we’ve used. For example, the argument

All male philosophers have beards.

Socrates is a male philosopher.

Therefore, Socrates has a beard.

is valid. If all male philosophers have beards, and if Socrates is a male philosopher, then it follows that he has a beard. If Socrates does not have a beard, then either he isn’t a male philosopher or...

• Chapter 9 Models for predicate logic
(pp. 128-148)

The language of the predicate calculus is not much use without a way of interpreting formulas in that language. We know how to express things in the language, but we have no way of testing argument forms or finding models to make formulas true or false. It is the task of this chapter to introduce models that enable us to do just that.

In the first half of the book, when we didn’t worry about predicates, names, variables and quantifiers, evaluations for our propositions were simple. You could find a ‘way things could be’ by assigning truth values to the...

• Chapter 10 Trees for predicate logic
(pp. 149-167)

Working with finite models is tedious – especially when you have three or more predicates, and you have to expand quantifiers with eight constants. And furthermore, the method isn’t going to be definitive with arguments containing any predicates of arity greater than 1. So, we need another method to deal with quantifiers. There is a method that works –trees.When we use trees to evaluate arguments, we construct the model as we go. We don’t have to decide in advance how many objects there are in the model. We can ‘introduce new objects’ as required.

There is a simple extension of...

• Chapter 11 Identity and functions
(pp. 168-182)

The language of predicate logic is all well and good as it stands – it is a vast improvement over prepositional logic – but there are still some things you cannot say. Say we have two names ‘Clark Kent’ and ‘Superman’, and we want to express the fact that they refer to the same person. We say it like this:

There is no way to say this simply in the language of first-order logic as we have it. TheisinClark Kent is Supermanis not the same sort of is as appears inClark Kent is a reporter.We cannot...

• Chapter 12 Definite descriptions
(pp. 183-191)

Names in our languages pick out objects. In our formal language, this job is done by names, which are completelyatomic(they have no significant parts) or are given by applyingfunction symbolsto other names. There is no other class of referring expressions in our language.

This seems like an odd fit. We have shown how for formulas, the meanings (interpretations) of complex formulas are made up out of the meanings of simpler formulas. It seems that things should work like this for referring expressions too. There seem to bedescriptionsthat pick out objects, in virtue of the...

• Chapter 13 Some things do not exist
(pp. 192-204)

According to the syllogistic logic of Aristotle, the following arguments are valid:

All tigers are dangerous.

So some tigers are dangerous.

No liars are honest.

So some liars are not honest.

Aristotle’s syllogistic logic dominated logical theory in Western civilisation for over 2,000 years. According to the predicate logic we have considered, both arguments have theinvalidforms

$(\forallx)\,(Fx\;\supsetGx)\;{\rm{therefore}}\;{\rm{(}}\exists{\rm{x)(}}Fx\;\&\;Gx)$

$\~(\existsx)(Fx\;\&\;Gx)\;{\rm{therefore}}\;{\rm{(}}\exists{\rm{x)(}}Fx\;\&\;\~Gx)$

The reason for the difference is straightforward. For syllogistic logic, an assumption was made that every category is inhabited. That is, every predicate has something in its extension. If all tigers are dangerous, pick one of the tigers. It is...

• Chapter 14 What is a predicate?
(pp. 205-211)

We will introduce the topic of this chapter by considering an argument.

Here is an argument, due to René Descartes, designed to show that reality is not merely material – there are some immaterial things too.

I can now imagine that my material body is not now existing.

I can not now imagine that I am not now existing.

Therefore, I am not now my material body.

This is an argument fordualism,the view that the universe contains immaterial things (souls, minds, gods, and things like that) as well as material things (bodies, tables, chairs, trees, and so on). Read...

• Chapter 15 What is logic?
(pp. 212-217)

Sometimes the answers you get with classical logic are odd, and not for reasons of vagueness or relevance, and not because of non-denoting terms, or opaque contexts. Here is an example:

The Prime Minister collects clocks.

Anyone who collects clocks has to be slightly mad.

We can use the dictionary

$Mx = x$is a Prime Minister

$Cx = x$collects clocks

$Sx = x$is slightly mad

$Px = x$is a person

to find the form. The argument has$(Ix)(Mx,Cx)$and$(\forall x)((Px\& Cx) \supset Sx))$as premises, and the conclusion is$(\exists x)(Px\& Sx)$. It is straightforward to show that it is invalid. One counterexample can...

7. Bibliography
(pp. 218-220)
8. Index
(pp. 221-225)